The first term of a G.P. is 1. The sum of the third term and fift…

Question

The first term of a G.P. is 1. The sum of the third term and fifth term is 90. Find the common ratio of G.P.

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Answer

Given: $a = 1$ and $a_3 + a_5 = 90$
$\Rightarrow ar^2 + ar^4 = 90$
$\Rightarrow a(r^2 + r4) = 90$
$\Rightarrow 1 \times \left( r ^ { 2 } + r ^ { 4 } \right) = 90$$\Rightarrow r^2 + r^4 = 90$
$\Rightarrow r^4 + r^2 - 90 = 0$
$\Rightarrow r ^ { 2 } = \frac { - 1 \pm \sqrt { ( 1 ) ^ { 2 } - 4 \times ( - 90 ) \times 1 } } { 2 \times 1 }$
$= \frac { - 1 \pm \sqrt { 1 + 360 } } { 2 } = \frac { - 1 \pm \sqrt { 361 } } { 2 }$
$= \frac { - 1 \pm 19 } { 2 }$=
$\Rightarrow r ^ { 2 } = \frac { - 1 + 19 } { 2 } = \frac { 18 } { 2 } = 9$or $r ^ { 2 } = \frac { - 1 - 19 } { 2 } = \frac { - 20 } { 2 } = - 10$ which is not possible
Therefore, the common ratio is $r = \pm 3$

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